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представительное множество. Оно может быть найдено путём попарного сравнения частотных классов системы K по отношению представительности с использованием леммы 3. Мощность системы K ограничена числом частотных классов с длиной слов не выше l, которое не превосходит где |A| —мощность алфавита A, а число пар классов из K не больше l2|A|. Поскольку |A| —константа, а трудоёмкость сравнения одной пары по представительности полиномиальна, процедура выделения минимального представительного множества полиномиальна по l. ■
ЛИТЕРАТУРА
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2. Шоломов Л. А. О функционалах, характеризующих сложность систем недоопределенных булевых функций // Проблемы кибернетики. Вып. 19. М.: Физматлит, 1967. С. 123-139.
3. Нечипорук Э. И. О сложности вентильных схем, реализующих булевские матрицы с неопределенными элементами // ДАН СССР. 1965. Т. 163. №1. С. 40-42.
4. Адельсон-Вельский Г. М., Диниц Е. А., Карзанов А. В. Потоковые алгоритмы. М.: Наука, 1975.
UDC 512.772.7 DOI 10.17223/2226308X/12/12
CHARACTERISTIC POLYNOMIALS OF THE CURVE y2 = x7 + ax4 + bx
OVER FINITE FIELDS
S. A. Novoselov1, Y. F. Boltnev
In this work, we list all possible characteristic polynomials of the Frobenius endomorphism for genus 3 hyperelliptic curves of type y2 = x7 + ax4 + bx over finite field Fq of characteristic p > 3.
Keywords: hyperelliptic curves, characteristic polynomials, point-counting, genus 3.
Introduction
Let Fq be a finite field of size q = pn, p > 2. In this note, we study the hyperelliptic curves of genus g = 3 of the form
C : y2 = x2g+1 + axg+1 + bx.
The Jacobian JC of the curves is split [1] over certain finite field extension:
JC ~ JDi X JD2,
where D1 and D2 are explicitly given curves. This fact allows us to reduce the problem of point-counting on the curve C to counting points on the curves D1 and D2.
For genus 2 case it was done in the works [2, 3]. The work [1] contains algorithms for g > 2 case. In this work, we give explicit formulae for the number of points on the Jacobian in the case of g = 3.
The point-counting on the curve is equivalent to finding of zeta-function of the curve
Z(C/Fq; T) = exp (£#C(i>)^) = (1 qT),
1The author is supported by RFBR according to the research project No. 18-31-00244.
Теоретические основы прикладной дискретной математики
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where Lo,q(T) = qgT2g + a^-1 T2g-1 + ... + agTg + afl_iTg-1 + ... + aiT + 1 and a, G Z,
la,| ^ ^qi/2fori = i,...,g-
Let Xc,q (T) be a characteristic polynomial of the Frobenius endomorphism. Then LC,q(T) = T2gxC,q(1/T) and #JC(Fq) = LCq(1) = xC,q(1). Therefore, the computation of #JC (Fq) is equivalent to the computation of the characteristic polynomial.
In this work, we enumerate all possible characteristic polynomials for the curve C in the case of g = 3.
1. Characteristic polynomials for genus 3 curves
Let C : y2 = x7 + ax4 + bx be a genus 3 hyperelliptic curve defined over a finite field Fq, q = pn, p > 3. Since, there is a map
(x,y) ^ (x3,xy)
from C to an elliptic curve E1 : y2 = x3 + ax2 + bx, we have
JC ~ E1 x A over Fq for some abelian surface A. Therefore,
XC,q (T) = XEi,q (T)XA,q (T) .
The characteristic polynomial for E1 can be efficiently computed using SEA-algorithm [4]. So, we only have to determine the coefficients of XA,q(T) = T4 — b1T3 + b2T2 — b1qT + q2. From [1, Th. 2], we have
Jc ~ E x Jd
over Fq [-^^j, where E2 is an elliptic curve with equation
y2 = x3 — 3 -^bx + a
and D is a hyperelliptic curve with equation
y2 = (x2 — 4\/b)(x3 — 3^/bx + a).
Moreover, the Jacobian JD is also split, since E1 ^ E2 in general. First we describe the characteristic polynomials in the simplest case when b is a cubic residue. In this case for each cubic root, we have a map to an elliptic curve, so we obtain the following theorem.
Theorem 1. Let C : y2 = x7 + ax4 + bx be a genus 3 hyperelliptic curve defined over a finite field Fq, q = pn, p > 3, and let b be a cubic residue. Then
1) if q = 1 (mod 6), then JC ~ E1 x over Fq and
XC,q (T) = (T2 — t1T + q)(T2 — t2T + q)2,
where E1 : y2 = x3 + ax2 + bx, E2 : y2 = x3 — 3-^^x + a are elliptic curves and t1, t2 are their traces of the Frobenius endomorphism;
2) if q = 5 (mod 6), then JC ~ E1 x E2 x _E2 over Fq and
XC,q (T) = (T2 — t1T + q)(T2 — t2T + q)(T2 + ^T + q), where E2 is a quadratic twist of E2.
In general case, we have JC ~ E1 x A, where A can be simple.
Theorem 2. Let C : y2 = x7 + ax4 + bx be a genus 3 hyperelliptic curve defined over a finite field Fq, q = pn, p > 3. Then
1) JC ~ E1 x A over Fq, where E1 is an elliptic curve with equation y2 = x3 + ax2 + bx and A is an abelian surface;
2) if q = 5 (mod 6), we have JC ~ E1 x E2 x E2 and
XC,q (T) = (T2 — t1T + q)(T2 — t2T + q)(T2 + ^T + q),
where E1,E2,t1,i2 are the same as in Theorem 1;
3) if q = 1 (mod 6) and ^b G Fq, then JC ~ E1 x over Fq and
XC,q (T) = (T2 — t1T + q)(T2 — t2T + q)2;
4) if q = 1 (mod 6), -^b G Fq and E2 is ordinary, then XC,q(T) = (T2 — t1T + q)XA(T), where XA(T) is one of the following polynomials:
• (T4 — t 2T3 + (¿2 — q)T2 — ¿2qT + q2), \ G Fq;
• (T4 + ¿2T3 + (¿2 — q)T2 + ¿2qT + q2), V G Fq;
• (T4 — 2^T3 + (¿2 + 2q)T2 — 2i2qT + q2), Vb G Fq, A is split;
• (t4 + 2t2T3 + (¿22 + 2q)T2 + 2t2qT + q2), Vb G Fq, A is split. Here, t2 is a trace of Frobenius of elliptic curve E2 : y2 = x3 — 3bx + ab;
5) if q = 1 (mod 6), ^b G Fq and E2 is supersingular, then A is supersingular and XC q(T) = (T2 — t1T + q)XA q(T) where XA q(T) is one of the following polynomials:
' • (T4 — qT2 + q2);
• (T4 + 2qT2 + q2);
• (T2 + q)(T ± \/q)2, p = 7 (mod 12), n is even, A is split;
• (T ± Vq)2, n is even, A is split;
• (T2 ± T Vq + q)2, n is even, A is simple;
• (T4 + \VqT3 + qT2 + q3/2T + q2), p = 1 (mod 5), n is even, A is simple;
• (T4 — \VqT3 + qT2 — q3/2T + q2), p = 1 (mod 10), n is even, A is simple.
Conclusion
In this work, we obtained the complete list of the characteristic polynomials for the genus 3 curve y2 = x7 + ax4 + bx in terms of traces of Frobenius of certain elliptic curves. Since #JC (Fq) = xc,q (T), this gives us the explicit formulae for the number of points on the Jacobian.
REFERENCES
1. Novoselov S. A. Counting Points on Hyperelliptic Curves of Type y2 = x2g+1 + ax9+1 + bx. https://arxiv.org/abs/1902.05992. 2019.
2. Satoh T. Generating genus two hyperelliptic curves over large characteristic finite fields. LNCS, 2009, vol. 5479, pp. 536-553.
3. Guillevic A. and Vergnaud D. Genus 2 hyperelliptic curve families with explicit Jacobian order evaluation and pairing-friendly constructions. LNCS, 2012, vol.7708, pp.234-253.
4. SchoofR. Counting points on elliptic curves over finite fields. J. de theorie des nombres de Bordeaux, 1995, vol.7, no. 1, pp. 219-254.
